An operator splitting finite element algorithm for transient as well as steady viscoelastic flows has been implemented using a special high level language Fasttalk. The momentum equation is solved by operator splitting with a preconditioned conjugate gradient method, and the constitutive equation is integrated by an implicit scheme with a consistent streamline upwind Petrov-Galerkin (SUPG) method. The high efficiency of the present decoupled scheme has enabled us to do mesh refinement study using more than 10(5) degrees of freedom on a workstation computer for the viscoelastic flow around a sphere in a tube, and a speed of CPU proportional to N-1.3 has been delivered by the current code (N is the total number of unknowns). Steady state solutions were obtained for Deborah numbers De less than or equal to 2.8 for the Upper-Convected Maxwell fluid. Comparison of the present results with those found in the literature (all due to coupled methods) shows excellent agreement on the predicted drag coefficient K for De less than or equal to 1.4. At higher De numbers (De greater than or equal to 1.6) the present decoupled computation predicted no levelling off in K, while the other coupled computations predicted a levelling off in K followed by an earlier loss of convergence.